Chapter 3 Context-Free Grammars, Context-Free Languages, Parse ...

60 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
Lemma 3.9.1 Given any context-free grammar G =(V,Σ,P,S) with m nonterminals A1,
..., Am, writingGas X = XH + K
as explained earlier, if GH is the grammar defined by the system of two matrix equations
X = KY + K,
Y = HY + H,
as explained above, then the components in X of the least-fixed points of the maps ΦG and
ΦGH are equal.
Proof . Let U be the least-fixed point of ΦG, andlet(V,W) be the least fixed-point of
ΦGH. We shall prove that U = V . For notational simplicity, let us denote Φ[U](H) asH[U]
and Φ[U](K) asK[U].
Since U is the least fixed-point of X = XH + K, wehave
U = UH[U]+K[U].
Since H[U] andK[U] do not contain any nonterminals, by a previous remark, K[U]H ∗ [U] is
the least-fixed point of X = XH[U]+K[U], and thus,
K[U]H ∗ [U] ≤ U.
On the other hand, by monotonicity,
K[U]H ∗
[U]H K[U]H ∗
[U] + K K[U]H ∗
[U]
and since U is the least fixed-point of X = XH + K,
U ≤ K[U]H ∗ [U].
≤ K[U]H ∗ [U]H[U]+K[U] =K[U]H ∗ [U],
Therefore, U = K[U]H ∗ [U]. We can prove in a similar manner that W = H[V ] + .
and
Let Z = H[U] + .Wehave
K[U]Z + K[U] =K[U]H[U] + + K[U] =K[U]H[U] ∗ = U,
H[U]Z + H[U] =H[U]H[U] + + H[U] =H[U] + = Z,
and since (V,W) is the least fixed-point of X = KY + K and Y = HY + H, wegetV≤ U
and W ≤ H[U] + .
We also have
V = K[V ]W + K[V ]=K[V ]H[V ] + + K[V ]=K[V ]H[V ] ∗ ,